In seven-dimensional

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a rectified 7-cube is a convex

uniform 7-polytope In seven-dimensional space, seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope Ridge (geometry), ridge being shared by exactly two 6-polytope Facet (mathematics), facets. A uniform 7-polytope is ...

, being a

rectification Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Recti ...

of the regular

7-cube In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol , being com ...

. There are unique 7 degrees of rectifications, the zeroth being the

, and the 6th and last being the

. Vertices of the rectified 7-cube are located at the edge-centers of the 7-ocube. Vertices of the birectified 7-cube are located in the square face centers of the 7-cube. Vertices of the trirectified 7-cube are located in the

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

cell centers of the 7-cube.

Rectified 7-cube

Alternate names

* rectified hepteract (Acronym rasa) (Jonathan Bowers)

Images

Cartesian coordinates

Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

for the vertices of a rectified 7-cube, centered at the origin, edge length

\sqrt\

are all permutations of: : (±1,±1,±1,±1,±1,±1,0)

Birectified 7-cube

Alternate names

* Birectified hepteract (Acronym bersa) (Jonathan Bowers)

Images

Cartesian coordinates

for the vertices of a birectified 7-cube, centered at the origin, edge length

\sqrt\

are all permutations of: : (±1,±1,±1,±1,±1,0,0)

Trirectified 7-cube

Alternate names

* Trirectified hepteract * Trirectified 7-orthoplex * Trirectified heptacross (Acronym sez) (Jonathan Bowers)Klitzing, (o3o3o3x3o3o4o - sez)

Images

Cartesian coordinates

for the vertices of a trirectified 7-cube, centered at the origin, edge length

\sqrt\

are all permutations of: : (±1,±1,±1,±1,0,0,0)

Related polytopes

Notes

References

H.S.M. Coxeter Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

: ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973 ** Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995
wiley.com
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380-407, MR 2,10*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559-591*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45* Norman Johnson ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. * o3o3o3x3o3o4o - sez, o3o3o3o3x3o4o - bersa, o3o3o3o3o3x4o - rasa

External links

Polytopes of Various Dimensions

{{polytopes 7-polytopes